12.2.1 - Hypothesis Testing

In testing the statistical significance of the relationship between two quantitative variables we will use the five step hypothesis testing procedure:

1. Check assumptions and write hypotheses

In order to use Pearson's \(r\) both variables must be quantitative and the relationship between \(x\) and \(y\) must be linear

Research Question Is the correlation in the population different from 0? Is the correlation in the population positive? Is the correlation in the population negative?
Null Hypothesis, \(H_{0}\) \(\rho=0\) \(\rho= 0\) \(\rho = 0\)
Alternative Hypothesis, \(H_{a}\) \(\rho \neq 0\) \(\rho > 0\) \(\rho< 0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate the test statistic

Minitab will not provide the test statistic for correlation. It will provide the sample statistic, \(r\), along with the p-value (for step 3).

Optional: If you are conducting a test by hand, a \(t\) test statistic is computed in step 2 using the following formula:

  \(t=\dfrac{r- \rho_{0}}{\sqrt{\dfrac{1-r^2}{n-2}}} \)

In step 3, a \(t\) distribution with \(df=n-2\) is used to obtain the p-value.

3. Determine the p-value

Minitab will give you the p-value for a two-tailed test (i.e., \(H_a: \rho \neq 0\)). If you are conducting a one-tailed test you will need to divide the p-value in the output by 2.

4. Make a decision

If \(p \leq \alpha\) reject the null hypothesis, there is evidence of a relationship in the population.

If \(p>\alpha\) fail to reject the null hypothesis, there is not enough evidence of a relationship in the population.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.